What Special About This Number
red for the properties in dozenal (dependent in which base is used, in this wiki, we always use the dozenal base). 0 is the additive identity. 1 is the multiplicative identity. 2 is the number of elements in the smallest field. 3 is the smallest number of sides that a simple (non-self-intersecting) polygon can have. 4 is the smallest number of colors sufficient to color all planar maps. 5 is the smallest k such that akxk+ak-1xk-1+...+a1x+a0 does not have algebraic solution. 6 is the smallest order of nonabelian group. 7 is the smallest k such that regular k-gon is not constructible using compass and straightedge. 8 is the smallest positive integer with no primitive roots. 9 is the only base such that all repunit numbers are triangular numbers. X is the smallest noncototient number. E is the smallest k such that regular k-gon is not constructible using neusis, or an angle trisector. 10 appears in the value of the Riemann zeta function at −1 (i.e. ζ(−1) = −1/10). 11 is the number of Archimedean solids. 12 is the smallest nontotient number. 13 is the smallest k>1 such that k-th cyclotomic polynomial has more terms than the largest prime factor of k. 14 is the only number of the form ab ≠ ba, with a, b nonnegative integers, a != b. 15 is the only positive Genocchi prime. 16 is the smallest known solitary number which is not coprime to its sum-of-divisors. 17 is the smallest number not appearing in the first 100 terms of Recamán's sequence (in fact, 17 is also the smallest number not appearing in the first 49000 terms of Recamán's sequence, the first time 17 appearing is term 49872, if 0 is term 0, 1 is term 1). 18 is the number of moves (quarter or half turns) required to optimally solve a Rubik's Cube in the worst case. 19 is the smallest number of distinct squares needed to tile a square. 1X is the numerator of an approximation of π (1X/7). 1E is the smallest number n such that the relative class number h- of cyclotomic field Q(zeta_n) is greater than 1. 20 is the largest number for which the Dirichlet characters are all real. 21 is the smallest square that can be written as a sum of 2 positive squares. 22 is the only positive number to be directly between a square and a cube. 23 is the number n'' for which (the largest number in the sequence of ''n for Collatz conjecture)/(n''2) is largest. (i.e. 5414/(23^2) = 10.7E7314) 24 is the smallest even number which is a (Fermat) pseudoprime to some nontrivial bases. 25 is the largest number ''n such that 2''x''2 + n'' is prime for all 0≤''x≤''n''−1. (since it is divisible by n'' for ''x = n'', one cannot do be better than this) 26 is the largest number with the property that all smaller numbers relatively prime to it are prime or 1. 27 is one of the only two numbers which is a repunit in three or more bases (not including base 1). 28 is the smallest number ''n such that the n''-th row of the modulo-2 Pascal's triangle (the top row, which contains only one 1, is the 0th row, not the 1st row), when read in binary, is not a number of the sides of constructible regular polygon. 29 is the largest number that is not a sum of distinct triangular numbers. 2X is the smallest number with the property that it and its neighbors have the same number of divisors. 2E is the smallest number whose reciprocal does not terminate and has even period length, but does not satisfy Midy's theorem. 30 is the smallest perfect power which is not a prime power. 31 is the smallest irregular prime. 32 is the magic number of the only non-trivial normal magic hexagon. 33 is the smallest ''n which is not power of 10 such that n''.''n.n''...''n.n''.1 (dot means concatenation) cannot be prime. 34 is the smallest ''n such that n''.111...111 (dot means concatenation) cannot be prime. 35 is the largest number ''n such that x''2 + ''x + n'' is prime for all 0≤''x≤''n''−2. (since it is divisible by n'' for ''x = n''−1, one cannot do be better than this) 36 is the largest number of sides of a regular polygon that can fill a point with other regular polygons. 37 is the smallest number ''n such that (define a(n): a(0)=a(1)=1; thereafter a(n+1) = sum(a(k)^2,k=0..n)/n) a(n) is not integer. 38 is the smallest n'' such that all of ''n.0, n''.1, ''n.2, ..., n''.E (dot means concatenation) are composite. (i.e. all of 10''n+0, 10''n''+1, 10''n''+2, ..., 10''n''+E are composite) 39 is the smallest odd positive integer that is not power of squarefree number. 3X is the largest even number which is a value of D for incrementally largest values of minimal x satisfying Pell equation x^2−Dy^2=1. 3E is the smallest base for which no generalized Wieferich primes are known. 40 is the largest number n'' such that the sum of the first ''n positive triangular numbers is also a triangular number. 41 is the smallest number with the property that it and its neighbors are not squarefree. 42 is the smallest number which can be written as the sum of of 2 squares in 2 different ways. 43 is the number of sides of a constructible polygon. 44 is the smallest untouchable number > 5 (the conjectured only odd untouchable number). 45 is the smallest prime that produces prime reciprocal magic square. 46 is the smallest totient number which is not totient of squarefree number. 47 is the largest triangular number in the Fibonacci sequence. 48 is the only number n such that no x^2 mod n is prime and n is not Euler's "numerus idoneus" (or convenient numbers, or idoneal numbers). 49 is the smallest number >1 of the form Φ''n''(2) which is neither prime nor Fermat pseudoprime base 2. 4X is the largest squarefree even number n'' such that the imaginary quadratic field Q(√−n) has class number 2. 4E is the smallest prime factor of the smallest composite Euclid number (i.e. 4E|(11#+1) = 15467 = 4E×365). 50 is the smallest order of nonsolvable group. 51 is conjectured to be the largest number ''n such that kn−1 and kn+1 are not both primes for all k'' ≤ 4''n. 52 is the smallest number that can be written as the sum of of 3 distinct squares in 2 ways. 53 is the largest number of the form a''n'' − b''n'' with no primitive prime factors (26 − 16). 54 is the smallest number >1 which is both square number and cube number. 55 is the smallest deceptive prime. 56 is the smallest k'' (and the only squarefree ''k) such that "n'' is deceptive prime" is equivalent to "''n is a weak pseudoprime base 10 and n'' is coprime to ''k". 57 is the smallest prime which is both Bernoulli irregular and Euler irregular. 58 is the smallest n'' which is not power of 10 and not congruent to 1 mod 11 (in which all such numbers are divisible by 11) such that (''nk''.1) (dot means concatenation) is composite for all 1≤''k≤1000. (the smallest k''≥1 such that this number is prime is 2781E5) 59 is the largest value of the smallest positive primitive root for all primes ''p ≤ 100000 (for p'' = 54201). 5X is the smallest weird number. 5E is the largest number whose square is one more than a factorial number. 60 is the smallest Achilles number. 61 is the largest squarefree number ''n such that the quadratic field OQ(√n) is a Euclidean domain. 62 is the number of different non-Hamiltonian polyhedra with a minimum number of vertices. 63 is the number of uniform polyhedra, excluding the infinite sets. 64 is the smallest n'' such that ''n-Fibonacci numbers cannot be primes. 65 is the largest number that cannot be written as a sum of distinct numbers whose reciprocals sum to 1. 66 is the dimension of the exceptional Lie group E''6. 67 is the smallest prime number ''p for which the real quadratic field Q√p has class number greater than 1. 68 is conjectured to be the largest possible number of consecutive integers n'' such that the quadratic polynomial ''an''2 + ''bn + c'' are primes (in the case ''n''2 + ''n + 35, which is prime for all −34≤''n''≤33, but not for n''=−35 or ''n = 34). 69 is an automorphic number. 6X is the largest number of protons for which stable nuclides exist. 6E is 70 is the smallest number n'' such that ''n is neither squarefree nor of the form p''a'q'b'' with p'', ''q primes, but no simple group with order n'' exists. 71 is the largest number ''n such that the sum of the first n'' positive square numbers is a triangular number. 76 is conjectured to be the largest base such that there is a nontrivial repunit which is also a nontrivial repunit in another base. 77 is the smallest positive integer expressible as a sum of two cubes in two different ways if negative roots are allowed. 83 is the smallest number whose factorial is greater than googol (=10100). X0 is the smallest number to appear 6 times in Pascal's triangle. X1 is the only perfect power which is a nontrivial repunit in some base (base 3). X8 is the largest number that is not a sum of distinct square numbers. XE is the smallest Sophie Germain prime which is irregular prime. E8 is the largest number whose square is a tetrahedral number. E9 is the smallest ''n>1 such that n''×2''n+1 is prime (Cullen prime). EE is the only product of twin primes which is not brilliant number. 100 is the largest square number in the Fibonacci sequence. 111 is the smallest irregular prime with irregular index greater than 1. 117 is the largest Heegner number. 12E is the smallest nonpalindromic number whose square is palindromic. 131 is the largest value x'' satisfying the Ramanujan–Nagell equation. 183 is the smallest Frugal number. 1E0 is the smallest number whose aliquot sequence has not yet been fully determined. 293 is the smallest Lucas-Carmichael number. 2X1 is conjectured to be the largest prime ''p whose smallest primitive root is larger than √p. 353 is the largest sum-product number. 3X8 is the smallest number which is a Rhonda number in some base (base 10). 3X9 is the smallest Carmichael number. 3XE is the largest known Wilson prime. 455 is the smallest prime which is a prime factor of a composite Fermat number. 497 is the first irregular prime to appear in the numerator of a Bernoulli number. 771 is the smallest Wieferich prime. 1001 is the smallest positive integer expressible as a sum of two cubes in two different ways. 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